The straight lines `7x-2y+10=0` and `7x+2y-10=0`form an isosceles triangle with the line `y=2.`The area of this triangle is equal to`(15)/7s qdotu n i t s` (b) `(10)/7s qdotu n i t s``(18)/7s qdotu n i t s`(d) none of these

Options

A. 15/7 sq. units
B. 10/7 sq. units
C. 18/7 sq. units
D. none of these

Explanation

To solve the problem, we need to find the area of the isosceles triangle formed by the lines \(7x - 2y + 10 = 0\), \(7x + 2y - 10 = 0\), and the line \(y = 2\).

### Step 1: Find the intersection points of the lines with \(y = 2\)

1. **Substituting \(y = 2\) in the first line:**
\[ 7x - 2(2) + 10 = 0 \implies 7x - 4 + 10 = 0 \implies 7x + 6 = 0 \implies 7x = -6 \implies x = -\frac{6}{7} \]
Thus, the point \(C\) is \((- \frac{6}{7}, 2)\).

2. **Substituting \(y = 2\) in the second line:**
\[ 7x + 2(2) - 10 = 0 \implies 7x + 4 - 10 = 0 \implies 7x - 6 = 0 \implies 7x = 6 \implies x = \frac{6}{7} \]
Thus, the point \(B\) is \((\frac{6}{7}, 2)\).

### Step 2: Find the length of the base \(BC\)

The length of the base \(BC\) can be calculated as:
\[ BC = \left| x_B - x_C \right| = \left| \frac{6}{7} - \left(-\frac{6}{7}\right) \right| = \left| \frac{6}{7} + \frac{6}{7} \right| = \left| \frac{12}{7} \right| = \frac{12}{7} \]

### Step 3: Find the height of the triangle

The height of the triangle can be determined by finding the vertical distance from the line \(y = 2\) to the vertex \(A\) of the triangle, which is the intersection of the two lines.

1. **Finding the intersection point \(A\) of the two lines:**
We can solve the equations:
\[ 7x - 2y + 10 = 0 \quad \text{(1)} \]
\[ 7x + 2y - 10 = 0 \quad \text{(2)} \]

Adding both equations:
\[ (7x - 2y + 10) + (7x + 2y - 10) = 0 \implies 14x = 0 \implies x = 0 \]

Substituting \(x = 0\) into equation (1):
\[ 7(0) - 2y + 10 = 0 \implies -2y + 10 = 0 \implies 2y = 10 \implies y = 5 \]
Thus, the point \(A\) is \((0, 5)\).

2. **Finding the height \(AD\):**
The height is the vertical distance from point \(A(0, 5)\) to the line \(y = 2\):
\[ AD = |y_A - y_{line}| = |5 - 2| = 3 \]

### Step 4: Calculate the area of the triangle

The area \(A\) of the triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{12}{7} \times 3 \]
Calculating this gives:
\[ \text{Area} = \frac{1}{2} \times \frac{12 \times 3}{7} = \frac{36}{14} = \frac{18}{7} \text{ square units} \]

### Conclusion
The area of the isosceles triangle formed by the given lines and the line \(y = 2\) is \(\frac{18}{7}\) square units.